On totally characteristic type nonlinear partial differential equations in the complex domain.

*(English)*Zbl 0961.35002The authors consider nonlinear singular partial differential equations \(t\partial_tu= F(t, x,\partial_xu)\). Their goal is to construct holomorphic solutions in a neighbourhood of the origin in \(\mathbb{C}^2\). The assumption \(F(0,x,0,0)\equiv 0\) implies that these solutions satisfy \(u(0,x)\equiv 0\) near \(x=0\). The right-hand side can be developed into the series
\[
F(t,x,u,v)= \alpha(x)t+ \beta(x)u+ \gamma(x) v+ \sum_{p+ q+\alpha\geq 2}a_{p, q,\alpha}(x) t^p u^q v^\alpha.
\]
Conditions with respect to \(\gamma= \gamma(x)\) determine the solvability behaviour. The authors are interested in the totally characteristic case, this means, \(\gamma(x)\not\equiv 0\) under the additional assumption \(\gamma(x)= xc(x)\) with \(c(0)\neq 0\). If a non-resonance condition of the form
\[
|i-\beta(0)- jc(0)|\geq \sigma(j+1)\quad \text{for any }(i,j)\in \mathbb{N}\times \mathbb{N}_0
\]
is satisfied for some positive \(\sigma\), then the starting equation has a unique holomorphic solution in a neighbourhood of the origin in \(\mathbb{C}^2\).

The non-resonance condition is useful for the construction of formal solutions. Their convergence is shown by the method of majorant power series. At the end of the paper the authors study a special class of higher-order totally characteristic partial differential equations. After defining a suitable symbol they formulate a kind of non-resonance condition for this symbol. This condition guarantees a unique holomorphic solution in a neighbourhood of the origin in \(\mathbb{C}^2\), too.

The non-resonance condition is useful for the construction of formal solutions. Their convergence is shown by the method of majorant power series. At the end of the paper the authors study a special class of higher-order totally characteristic partial differential equations. After defining a suitable symbol they formulate a kind of non-resonance condition for this symbol. This condition guarantees a unique holomorphic solution in a neighbourhood of the origin in \(\mathbb{C}^2\), too.

Reviewer: M.Reissig (Freiberg)

##### MSC:

35A07 | Local existence and uniqueness theorems (PDE) (MSC2000) |

35C10 | Series solutions to PDEs |

35A10 | Cauchy-Kovalevskaya theorems |

35A20 | Analyticity in context of PDEs |

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\textit{H. Chen} and \textit{H. Tahara}, Publ. Res. Inst. Math. Sci. 35, No. 4, 621--636 (1999; Zbl 0961.35002)

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